We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot H^{ 1-\tau}(\mathbb{ R}^2) \cap \dot H^s(\mathbb{ R}^2)$ with $s > 3$ and for any $0 < \tau <1$. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to $H^{ 20}(\mathbb{ R}^2)$. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in $H^{ 1-\tau}(\mathbb{ R}^2) \cap \dot H^s(\mathbb{ R}^2)$ with $s > 3$ and $0 < \tau <1$. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity $\|u_2(t)\|_{ L^\infty (\mathbb{ R}^2)}$ for initial data only in $\dot H^{ 1-\tau}(\mathbb{ R}^2) \cap \dot H^s(\mathbb{ R}^2)$ with $s >3$.
Categories: math.AP math-ph math.MP